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Minimising Weighted Mean Distortion
A thesis presented in partial fulfilment of the requirements
for the degree of
Doctor of Philosophy
in
Mathematics
at Massey University, Albany, New Zealand
Maarten Nicolaas Mc Kubre-Jordens
2009
ABSTRACT
There has been considerable recent interest in geometric function theory,
nonlinear partial differential equations, harmonic mappings, and the connection of these to minimal energy phenomena. This work explores Nitsche’s
1962 conjecture concerning the nonexistence of harmonic mappings between
planar annuli, cast in terms of distortion functionals. The connection between the Nitsche problem and the famous Grötzsch problem is established
by means of a weight function. Traditionally, these kinds of problems are
investigated in the class of quasiconformal mappings, and the assumption is
usually made a priori that solutions preserve various symmetries. Here the
conjecture is solved in the much wider class of mappings of finite distortion,
symmetry-preservation is proved, and ellipticity of the variational equations
concerning these sorts of general problems is established. Furthermore, various alternative interpretations of the weight function introduced herein lead
to an interesting analysis of a much wider variety of critical phenomena —
when the weight function is interpreted as a thickness, density or metric, the
results lead to a possible model for tearing or breaking phenomena in material science. These physically relevant critical phenomena arise, surprisingly,
out of purely theoretical considerations.
iv
ACKNOWLEDGEMENTS
First and foremost, my thanks to my supervisor Gaven Martin, for his valuable and sage guidance and support. This work is an ode to his skill as
supervisor.
To the people who have helped me throughout my study by contributing
their time and knowledge — my co-supervisor Carlo Laing, Graeme Wake,
Winston Sweatman, Shaun Cooper, Chris Tuffley, and many others — thank
you.
I am grateful also to the various institutes who have hosted me, funded
me, or otherwise contributed. These are: the New Zealand Institute for
Mathematics and its Applications, who have generously funded my study;
Massey University’s Institute of Information and Mathematical Sciences, who
have hosted me, providing space, computer equipment, access to Massey
University’s resources, and support for conferences; and the New Zealand
Institute for Advanced Study, which has provided support in many ways.
I am especially grateful and indebted to the love of my life, my wife
Alexandra, for her tireless dedication and support, and for the mammoth
task of proofreading at the end. Special thanks to my fellow postgraduate
students, who had the tenacity both to put up with me and to let me put up
with them.
The invaluable support I have received from a much longer list of people,
academic and otherwise, should not go unmentioned. Those who I have
acknowledged here by name are but the tip of the proverbial iceberg. A
thesis takes much more than talent or hard work: it takes supporters of all
kinds.
Dedicated to the memory of Jan Vermeulen, my Opa.
24 September 1932 – 21 September 2009
vi
CONTENTS
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
A brief overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.1
Holomorphic functions . . . . . . . . . . . . . . . . . . . . . .
6
1.2
Conformal mappings & Riemann’s Mapping Theorem . . . . .
8
1.2.1
Normal Families . . . . . . . . . . . . . . . . . . . . . .
9
1.2.2
Riemann’s Mapping Theorem . . . . . . . . . . . . . .
9
1.3
Sobolev spaces
. . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4
Quasiconformal mappings & moduli of curve families . . . . . 12
1.4.1
Modulus of an annulus . . . . . . . . . . . . . . . . . . 14
1.4.2
Quasiconformality, geometrically and analytically. . . . 17
1.5
Mappings of finite distortion & distortion functions . . . . . . 19
1.6
The calculus of variations . . . . . . . . . . . . . . . . . . . . 22
1.6.1
1.7
Multiple independent variables
. . . . . . . . . . . . . 25
Harmonic mappings & Dirichlet’s principle . . . . . . . . . . . 27
2. Minimisation Problems . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1
Conformal mapping problems . . . . . . . . . . . . . . . . . . 30
2.2
Quasiconformal mapping problems . . . . . . . . . . . . . . . 31
2.3
Mappings of finite distortion . . . . . . . . . . . . . . . . . . . 33
2.4
The Teichmüller problem . . . . . . . . . . . . . . . . . . . . . 34
2.5
Nitsche-type problems & Nitsche’s conjecture . . . . . . . . . 35
2.6
Grötszch-type problems
. . . . . . . . . . . . . . . . . . . . . 37
viii
2.7
Equivalence of Nitsche- and Grötszch-type problems . . . . . . 38
3. Minimisers of Distortion Functionals . . . . . . . . . . . . . . . . . 41
3.1
The symmetry assumption for the Grötzsch problem . . . . . 41
3.2
A key inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3
The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1
3.4
The theorem for the annulus . . . . . . . . . . . . . . . 52
Critical Nitsche-type phenomena . . . . . . . . . . . . . . . . 53
3.4.1
The Nitsche phenomenon . . . . . . . . . . . . . . . . . 53
3.4.2
More generality for Φ0 = 0 . . . . . . . . . . . . . . . . 54
3.4.3
Failure of Nitsche phenomenon: Φ0 unbounded . . . . . 57
3.4.4
Borderline case: Φ0 bounded . . . . . . . . . . . . . . . 58
4. Ellipticity of the Euler-Lagrange variational equations . . . . . . . . 63
4.1
Variational equations . . . . . . . . . . . . . . . . . . . . . . . 63
4.2
The initial result . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3
The weighted expansion . . . . . . . . . . . . . . . . . . . . . 71
5. Interpreting the Weight Function — Specific Cases & Applications . 75
5.1
Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2
Metric interpretation . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.1
Solving the metric equation . . . . . . . . . . . . . . . 80
5.2.2
Euclidean (flat) metrics
5.2.3
Spherical metrics . . . . . . . . . . . . . . . . . . . . . 82
5.2.4
Hyperbolic metrics . . . . . . . . . . . . . . . . . . . . 85
5.2.5
Constant nonzero curvature . . . . . . . . . . . . . . . 85
5.2.6
Nonconstant curvature . . . . . . . . . . . . . . . . . . 87
. . . . . . . . . . . . . . . . . 81
6. Sequitur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1
Further research . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendix
95
A. Common definitions and notation . . . . . . . . . . . . . . . . . . . 97
ix
B. Selected background theorems and propositions . . . . . . . . . . . 103
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
x
LIST OF FIGURES
1.1
1.2
1.3
The Riemann Mapping Theorem. . . . . . . . . . . . . . . . . 10
A curve family for the planar annulus. . . . . . . . . . . . . . 15
Local action of a quasiconformal map. . . . . . . . . . . . . . 17
2.1
2.2
2.3
2.4
Deformations in the plane (left) and the flat metric (right). .
A doubly connected region is conformally equivalent to a circular annulus. . . . . . . . . . . . . . . . . . . . . . . . . . .
The Teichmüller problem. . . . . . . . . . . . . . . . . . . .
Equivalence of Nitsche- and Grötzsch-type problems. . . . .
3.1
Beyond the Nitsche bound. . . . . . . . . . . . . . . . . . . . . 57
5.1
5.2
5.3
5.4
5.5
Stretching of a cut of base length ` (α = 12 ). . . . . . . . . . .
Stretching of a block with an open cut (α = 21 ). . . . . . . . .
Stretching of a block with a straight-line cut (α = 12 ). . . . . .
Stretching of a cusp (α = 1, n = 2 (top), n = 5 (bottom)). . .
Spherical (left), flat (center), and hyperbolic (right) cylinders.
. 29
. 31
. 34
. 39
77
77
78
78
90
xii